You can solve this since you can develop 3 equations for your 3 unknowns. Each equation is made using the Pythagorean theorem:
1) x^2 = y^2 + 16^2
2) z^2 = y^2 + 4^2
3) (16 + 4)^2 = z^2 + x^2
Simplifying the above we get:
1) x^2 = y^2 + 256
2) z^2 = y^2 + 16
3) 400 = z^2 + x^2
We want to solve for z. There are several ways but an easy one is with substitution. Start by rearranging 2) and 3) like this:
2’) y^2 = z^2 - 16
3’) x^2 = 400 - z^2
Now we have expressed x and y in terms of z. Plug these values into the unused equation, 1), to solve for z:
1’) 400 - z^2 = z^2 - 16 + 256
2z^2 = 160
z^2 = 80
z = √80
z = √(16)(5)
z = 4√5